For a complex number z=a+biz=a+bi, we can rewrite it in the form z=r(costheta+isintheta)z=r(cosθ+isinθ), where r=sqrt(a^2+b^2)r=√a2+b2 and theta=tan^(-1)(b/a)θ=tan−1(ba)
z_1=2+iz1=2+i
r_1=sqrt(2^2+1^2)=sqrt(5)r1=√22+12=√5
theta_1=tan^(-1)(1/2)θ1=tan−1(12)
z_1=sqrt(5)(cos(tan^(-1)(1/2))+isin(tan^(-1)(1/2))z1=√5(cos(tan−1(12))+isin(tan−1(12))
z_2=3+7iz2=3+7i
r_2=sqrt(3^2+7^2)=sqrt(58)r2=√32+72=√58
theta_2=tan^(-1)(7/3)θ2=tan−1(73)
z_2=sqrt(5)(cos(tan^(-1)(7/3))+isin(tan^(-1)(7/3))z2=√5(cos(tan−1(73))+isin(tan−1(73))
z_1/z_2=r_1/r_2(cos(theta_1-theta_2)+isin(theta_1-theta_2))z1z2=r1r2(cos(θ1−θ2)+isin(θ1−θ2))
z_1/z_2=sqrt(5)/sqrt(58)(cos(tan^(-1)(1/2)-tan^(-1)(7/3))+isin(tan^(-1)(1/2)-tan^(-1)(7/3)))z1z2=√5√58(cos(tan−1(12)−tan−1(73))+isin(tan−1(12)−tan−1(73)))
~~sqrt(290)/58(cos(-40.24)+isin(-40.24))≈√29058(cos(−40.24)+isin(−40.24))
Since cos(x)=cos(-x)cos(x)=cos(−x) and sin(-x)=-sin(x)sin(−x)=−sin(x)
=sqrt(290)/58(cos(40.24)-isin(40.24))=√29058(cos(40.24)−isin(40.24))
